Avalanche dynamics of crackle sound in the lung

Avalanche-size distribution of Cayley tree

Attacks on networks is a very important issue in developing strategies of eradicating spreads of malicious phenomena in networks, such as epidemics and fake information. This field of research is referred to as networks immunization. The traditional approach to evaluating the effectiveness of attacks on networks focuses on measuring macro parameters related to the entire attack, such as the critical probability of a percolation occurrence in the network [Formula: see text] and the relative size of the largest component in the network, known as the giant component, but not considering the attack on a micro perspective, which is the analysis of node removals, during an attack, themselves, their characteristics and results. In this paper we present and apply the last method of focusing on the micro scale of an attack. Based on the theory of percolation in networks, we analyze the phenomenon of an avalanche which results due to a single node removal from a network. An avalanche is a state in which a removal of a single node from the giant component of a network leads to the disconnection of additional nodes. This process significantly contributes to the fragmentation (immunization) of the network, comparing to the impact of the initial node removal alone. Specifically, we focus on the size parameter of an avalanche, which is the number of nodes that are disconnected from the giant component due to a single node removal. Relating to a random attack on a network of the type of Cayley tree, we derive analytically the distribution of the sizes of avalanches that occur during the entire attack on it, until the network is dismantled (immunized) and the attack is terminated.

Modeling Lung Derecruitment in VILI Due to Fluid-Occlusion: The Role of Emergent Behavior

Ventilator-induced lung injury (VILI) is driven by the processes of volutrauma and atelectrauma, which can act synergistically to compromise the blood-gas barrier. We have postulated that this synergy arises through a rich-get-richer mechanism whereby atelectrauma causes holes to form in the blood-gas barrier while concomitant volutrauma causes susceptible holes to progressively enlarge as VILI worsens. We previously developed an analytical model based on this idea that accurately predicts the progressive increases in lung elastance seen immediately following a recruitment maneuver as VILI progresses over the course of hours. In the present study we extend this model to account for the rate of change of elastance, due to closure of lung units, in the minutes following a recruitment maneuver. We found that the distribution of unit closing velocities throughout the lung can be described by a power law with an exponent of -2 that matches previously published power laws associated with the dynamics of lung recruitment. Our model thus reveals lung collapse as an example of emergent complex behavior and links the dynamics of altered function in the injured lung to structural damage in a way that explains the mechanisms of injury progression arising from the ongoing stresses and strains applied by mechanical ventilation.

Instantaneous frequency based index to characterize respiratory crackles

BACKGROUND

Crackle is a lung sound widely employed by health staff to identify respiratory diseases. The two-cycle duration (2CD) is a quantitative index pointed out by the American Thoracic Society and the European Respiratory Society to classify respiratory crackles as fine or coarse. However, this index, measured in the time domain, is highly affected by noise and filters of recording systems. Such factors hamper the analysis of data reported by different research groups. This work proposes a new index based on the instantaneous frequency of crackles estimated by means of discrete-time pseudo Wigner-Ville distribution.

METHOD

Comparisons between 2CD and the proposed index were carried out for simulated and actual crackles. Normal breathing sounds were added to simulated crackles; the resulting signals were then applied to a band-pass filter that mimics those belonging to lung sound acquisition systems. Thus, the impact of noise and filtering on these two indices was assessed for simulated crackles. Kruskal-Wallis and Dunn's tests as well as Gaussian mixture model (GMM) were applied to the two indices measured from 382 actual crackles belonging to open databases.

RESULTS

The proposed index is much less susceptible to waveform distortions due to noise and filtering when compared to the 2CD. Thus, the statistical analyses allow the identification of two classes of crackles from actual databases; the same does not occur when using 2CD.

CONCLUSIONS

The new proposed index has the potential to contribute for a better characterization of crackles generated by different respiratory diseases, assisting their diagnosis during clinical exams.

Non-equilibrium cytoquake dynamics in cytoskeletal remodeling and stabilization

The cytoskeleton (CSK) is a tensed fiber framework that supports, shapes and stabilizes the cell. The CSK is in a constant state of remodeling, moreover, which is an active non-equilibrium thermodynamic process. We report here that cytoskeletal remodeling involves reconfigurations that are not only sudden but also are transmitted to great distances within the cell in a fashion reminiscent of quakes in the Earth's crust. Remarkably, these events in the cell conform both qualitatively and quantitatively to empirical laws typical of earthquakes, including hierarchical fault structures, cumulative energy distributions following the Gutenberg-Richter law, and rate of after-shocks following Omori's law. While it is well-established that remodeling and stabilization of the cytoskeleton are non-equilibrium process, these new unanticipated observations establish that these processes are also remarkably non-local and strongly cooperative.

Analytical solution for a class of network dynamics with mechanical and financial applications

We show that for a certain class of dynamics at the nodes the response of a network of any topology to arbitrary inputs is defined in a simple way by its response to a monotone input. The nodes may have either a discrete or continuous set of states and there is no limit on the complexity of the network. The results provide both an efficient numerical method and the potential for accurate analytic approximation of the dynamics on such networks. As illustrative applications, we introduce a quasistatic mechanical model with objects interacting via frictional forces and a financial market model with avalanches and critical behavior that are generated by momentum trading strategies.

Complexity and emergent phenomena

Complex biological systems operate under non-equilibrium conditions and exhibit emergent properties associated with correlated spatial and temporal structures. These properties may be individually unpredictable, but tend to be governed by power-law probability distributions and/or correlation. This article reviews the concepts that are invoked in the treatment of complex systems through a wide range of respiratory-related examples. Following a brief historical overview, some of the tools to characterize structural variabilities and temporal fluctuations associated with complex systems are introduced. By invoking the concept of percolation, the notion of multiscale behavior and related modeling issues are discussed. Spatial complexity is then examined in the airway and parenchymal structures with implications for gas exchange followed by a short glimpse of complexity at the cellular and subcellular network levels. Variability and complexity in the time domain are then reviewed in relation to temporal fluctuations in airway function. Next, an attempt is given to link spatial and temporal complexities through examples of airway opening and lung tissue viscoelasticity. Specific examples of possible and more direct clinical implications are also offered through examples of optimal future treatment of fibrosis, exacerbation risk prediction in asthma, and a novel method in mechanical ventilation. Finally, the potential role of the science of complexity in the future of physiology, biology, and medicine is discussed.

Airway reopening through catastrophic events in a hierarchical network

When you reach with your straw for the final drops of a milkshake, the liquid forms a train of plugs that flow slowly initially because of the high viscosity. They then suddenly rupture and are replaced with a rapid airflow with the characteristic slurping sound. Trains of liquid plugs also are observed in complex geometries, such as porous media during petroleum extraction, in microfluidic two-phase flows, or in flows in the pulmonary airway tree under pathological conditions. The dynamics of rupture events in these geometries play the dominant role in the spatial distribution of the flow and in determining how much of the medium remains occluded. Here we show that the flow of a train of plugs in a straight channel is always unstable to breaking through a cascade of ruptures. Collective effects considerably modify the rupture dynamics of plug trains: Interactions among nearest neighbors take place through the wetting films and slow down the cascade, whereas global interactions, through the total resistance to flow of the train, accelerate the dynamics after each plug rupture. In a branching tree of microchannels, similar cascades occur along paths that connect the input to a particular output. This divides the initial tree into several independent subnetworks, which then evolve independently of one another. The spatiotemporal distribution of the cascades is random, owing to strong sensitivity to the plug divisions at the bifurcations.

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T{c} to be the time for which the market volatility V(T{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws-the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Modeling the complex dynamics of derecruitment in the lung

Recruitment maneuvers using deep inflations (DI) have long been used clinically with the objective of recruiting collapsed regions of the lung. Considerable uncertainty continues to exist, however, as to how best to employ recruitment maneuvers or even if they should be used routinely at all for patients receiving mechanical ventilation. Much of this uncertainty may arise from a lack of understanding about the dynamic nature of recruitment and derecruitment. To shed some light on this complex issue, we developed a time-dependent computational model of recruitment and derecruitment in the lung based on a symmetrically bifurcating airway tree in which each branch has a critical closing and opening pressure as well as pressure-dependent opening and closing speeds. Starting from the fully open state, the model underwent regular ventilation for 8 min followed by a series of identical DIs separated by 5 min of identical regular ventilation. We found that the geographical nature and extent of derecruitment before and 5 min after each DI were not always the same, demonstrating that the model exhibits multiple stable states. We conclude that the effectiveness of a recruitment maneuver is not only simply a function of the duration and magnitude of a DI, but may also have an unpredictable component arising from the distributed bi-stable nature of the derecruitment process itself.

Estimating the diameter of airways susceptible for collapse using crackle sound

Airways that collapse during deflation generate a crackle sound when they reopen during subsequent reinflation. Since each crackle is associated with the reopening of a collapsed airway, the likelihood of an airway to be a crackle source is identical to its vulnerability to collapse. To investigate this vulnerability of airways to collapse, crackles were recorded during the first inflation of six excised rabbit lungs from the collapsed state, and subsequent reinflations from 5, 2, 1, and 0 cmH(2)O end-expiratory pressure levels. We derived a relationship between the amplitude of a crackle sound at the trachea and the generation number (n) of the source airway where the crackle was generated. Using an asymmetrical tree model of the rabbit airways with elastic walls, airway vulnerability to collapse was also determined in terms of airway diameter D. During the reinflation from end-expiratory pressure = 0 cmH(2)O, the most vulnerable airways were estimated to be centered at n = 12 with a peak. Vulnerability in terms of D ranged between 0.1 and 1.3 mm, with a peak at 0.3 mm. During the inflation from the collapsed state, however, vulnerability was much less localized to a particular n or D, with maximum values of n = 8 and D = 0.75 mm. Numerical simulations using a tree model that incorporates airway opening and closing support these conclusions. Thus our results indicate that there are airways of a given range of diameters that can become unstable during deflation and vulnerable to collapse and subsequent injury.

The role of time and pressure on alveolar recruitment

Inappropriate mechanical ventilation in patients with acute respiratory distress syndrome can lead to ventilator-induced lung injury (VILI) and increase the morbidity and mortality. Reopening collapsed lung units may significantly reduce VILI, but the mechanisms governing lung recruitment are unclear. We thus investigated the dynamics of lung recruitment at the alveolar level. Rats (n = 6) were anesthetized and mechanically ventilated. The lungs were then lavaged with saline to simulate acute respiratory distress syndrome (ARDS). A left thoracotomy was performed, and an in vivo microscope was placed on the lung surface. The lung was recruited to three recruitment pressures (RP) of 20, 30, or 40 cmH(2)O for 40 s while subpleural alveoli were continuously filmed. Following measurement of microscopic alveolar recruitment, the lungs were excised, and macroscopic gross lung recruitment was digitally filmed. Recruitment was quantified by computer image analysis, and data were interpreted using a mathematical model. The majority of alveolar recruitment (78.3 +/- 7.4 and 84.6 +/- 5.1%) occurred in the first 2 s (T2) following application of RP 30 and 40, respectively. Only 51.9 +/- 5.4% of the microscopic field was recruited by T2 with RP 20. There was limited recruitment from T2 to T40 at all RPs. The majority of gross lung recruitment also occurred by T2 with gradual recruitment to T40. The data were accurately predicted by a mathematical model incorporating the effects of both pressure and time. Alveolar recruitment is determined by the magnitude of recruiting pressure and length of time pressure is applied, a concept supported by our mathematical model. Such a temporal dependence of alveolar recruitment needs to be considered when recruitment maneuvers for clinical application are designed.

Dynamic scaling approach to study time series fluctuations

We propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface-growth problem. In this framework, the fluctuation sampling time interval plays the role of time variable, whereas the physical time is treated as the analog of spatial variable. In this way we found that the fluctuations of many real-world time series satisfy the analog of the Family-Viscek dynamic scaling ansatz. This finding permits us to use the powerful tools of kinetic roughening theory to classify, model, and forecast the fluctuations of real-world time series.

Interpretation of spirometry through signal analysis

Chronic obstructive pulmonary disease is attaining alarming proportions that requires more objective and quantitative ways for the diagnosis and the evaluation / stratification of, both, the disease and the therapeutic outcomes. Within this context, the present study explores the possibility to increase the effectiveness of spirometry through signal analysis. Expiratory flow results from converging airflows at different levels of airway branching. Furthermore, along a branching network of air conduits, the characteristics of converging air currents determine those of the resulting air flow. Thus, for the human bronchial tree, the characteristics of air currents within the smaller branches are, ideally, conserved at the expiratory flow recorded at the mouth. This makes it theoretically possible to use signal analysis methodologies in order to identify the characteristics of airflow along the different levels of the respiratory tree. The present study reports on an attempt to identify alterations non-invasively in the frequency spectrum of the first derivative of the Forced Vital Capacity curve of patients presenting with different respiratory conditions. Such alterations can be attributed to the onset and operation of the airway closure phenomenon that limits airflow, during forced expiration. Fundamental to the design of the study was the notion that the forced expiratory output of the respiratory system is determined by the bronchial tree and the upper respiratory tract. These two entities shape the air flow that is expelled from the collective airspace of the bronchial tree subdivisions distal to the terminal bronchi. At the end we were able to identify simple measures that are derived from the power spectrum of the derivative of the spirometric curve that permit the definition of specific filters and allow for the accurate classification of, at least, the basic types of respiratory disease.

Monte Carlo simulation of liquid bridge rupture: application to lung physiology

In the course of certain lung diseases, the surface properties and the amount of fluids coating the airways changes and liquid bridges may form in the small airways blocking the flow of air, impairing gas exchange. During inhalation, these liquid bridges may rupture due to mechanical instability and emit a discrete sound event called pulmonary crackle, which can be heard using a simple stethoscope. We hypothesize that this sound is a result of the acoustical release of energy that had been stored in the surface of liquid bridges prior to its rupture. We develop a lattice gas model capable of describing these phenomena. As a step toward modeling this process, we address a simpler but related problem, that of a liquid bridge between two planar surfaces. This problem has been analytically solved and we use this solution as a validation of the lattice gas model of the liquid bridge rupture. Specifically, we determine the surface free energy and critical stability conditions in a system containing a liquid bridge of volume Omega formed between two parallel planes, separated by a distance 2h, with a contact angle Theta using both Monte Carlo simulation of a lattice gas model and variational calculus based on minimization of the surface area with the volume and the contact angle constraints. In order to simulate systems with different contact angles, we vary the parameters between the constitutive elements of the lattice gas. We numerically and analytically determine the phase diagram of the system as a function of the dimensionless parameters hOmega(-1/3) and Theta. The regions of this phase diagram correspond to the mechanical stability and thermodynamical stability of the liquid bridge. We also determine the conditions for the symmetrical versus asymmetrical rupture of the bridge. We numerically and analytically compute the release of free energy during rupture. The simulation results are in agreement with the analytical solution. Furthermore, we discuss the results in connection to the rupture of similar bridges that exist in diseased lungs.

Cross-correlation of instantaneous phase increments in pressure-flow fluctuations: applications to cerebral autoregulation

We investigate the relationship between the blood flow velocities (BFV) in the middle cerebral arteries and beat-to-beat blood pressure (BP) recorded from a finger in healthy and post-stroke subjects during the quasisteady state after perturbation for four different physiologic conditions: supine rest, head-up tilt, hyperventilation, and CO2 rebreathing in upright position. To evaluate whether instantaneous BP changes in the steady state are coupled with instantaneous changes in the BFV, we compare dynamical patterns in the instantaneous phases of these signals, obtained from the Hilbert transform, as a function of time. We find that in post-stroke subjects the instantaneous phase increments of BP and BFV exhibit well-pronounced patterns that remain stable in time for all four physiologic conditions, while in healthy subjects these patterns are different, less pronounced, and more variable. We propose an approach based on the cross-correlation of the instantaneous phase increments to quantify the coupling between BP and BFV signals. We find that the maximum correlation strength is different for the two groups and for the different conditions. For healthy subjects the amplitude of the cross-correlation between the instantaneous phase increments of BP and BFV is small and attenuates within 3-5 heartbeats. In contrast, for post-stroke subjects, this amplitude is significantly larger and cross-correlations persist up to 20 heartbeats. Further, we show that the instantaneous phase increments of BP and BFV are cross-correlated even within a single heartbeat cycle. We compare the results of our approach with three complementary methods: direct BP-BFV cross-correlation, transfer function analysis, and phase synchronization analysis. Our findings provide insight into the mechanism of cerebral vascular control in healthy subjects, suggesting that this control mechanism may involve rapid adjustments (within a heartbeat) of the cerebral vessels, so that BFV remains steady in response to changes in peripheral BP.

Biofluid Mechanics of the Pulmonary System

Presents an overview of leading areas of discovery in bio-fluid mechanics related to the pulmonary system, with particular reference to the airways. Areas briefly reviewed include airway gas dynamics, impedance studies, collapsible-tube studies, and airway liquid studies. Emphasis is placed on promising further directions, such as analysis of interacting fluid-mechanical or fluid-structure phenomena, multi-scale modeling across widely varying length and time scales, and integration of advanced simulations into respiratory investigation and pulmonary medicine.

Perimeter growth of a branched structure: application to crackle sounds in the lung

We study an invasion percolation process on Cayley trees and find that the dynamics of perimeter growth is strongly dependent on the nature of the invasion process, as well as on the underlying tree structure. We apply this process to model the inflation of the lung in the airway tree, where crackling sounds are generated when airways open. We define the perimeter as the interface between the closed and opened regions of the lung. In this context we find that the distribution of time intervals between consecutive openings is a power law with an exponent beta approximately 2. We generalize the binary structure of the lung to a Cayley tree with a coordination number Z between 2 and 4. For Z=4, beta remains close to 2, while for a chain, Z=2 and beta=1, exactly. We also find a mean field solution of the model.

Fluid transport in branched structures with temporary closures: a model for quasistatic lung inflation

We analyze the problem of fluid transport through a model system relevant to the inflation of a mammalian lung, an asymmetric bifurcating structure containing random blockages that can be removed by the pressure of the fluid itself. We obtain a comprehensive description of the fluid flow in terms of the topology of the structure and the mechanisms which open the blockages. We show that when calculating averaged flow properties of the fluid, the tree structure can be partitioned into a linear superposition of one-dimensional chains. In particular, we relate the pressure-volume P-V relationship of the fluid to the distribution Pi(n) of the generation number n of the tree's terminal branches, a structural property. We invert this relation to obtain a statistical description of the underlying branching structure of the lung, by analyzing experimental pressure-volume data from dog lungs. The Pi(n) extracted from the experimental P-V data agrees well with available data on lung branching structure. Our general results are applicable to any physical system involving transport in bifurcating structures with removable closures.

Finite-size effects of avalanche dynamics

We study the avalanche dynamics of a system of globally coupled threshold elements receiving random input. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrary system sizes N using geometrical arguments in the system's configuration space. For finite systems, approximate power-law behavior is obtained in the nonconservative regime, whereas for N--> infinity, critical behavior with an exponent of -3/2 is found in the conservative case only. We compare these results to the avalanche properties found in networks of integrate-and-fire neurons, and relate the different dynamical regimes to the emergence of synchronization with and without oscillatory components.

Correlation properties of tidal volume and end-tidal O2 and CO2 concentrations in healthy infants

We investigated whether breath-to-breath fluctuations in tidal volume (VT) and end-tidal O2 and CO2 exhibit long-range correlations and whether parameters describing the correlations can be used as noninvasive descriptors of control of breathing. We measured VT and end-tidal O2 and CO2 over n = 352 +/- 104 breaths in 26 term, healthy, unsedated infants (mean age +/- SD: 36 +/- 6 days) and calculated the detrended fluctuation function [F(n)]. The F(n) of the breath-to-breath time series of VT, O2, and CO2 revealed a linear increase with a breath number on log-log plots with a slope that was significantly different from 0.5 (random) and thus consistent with scale-invariant behavior. Long-range correlations were stronger for O2 than for VT and CO2. The F(n) of many individual signals exhibited a crossover behavior indicating that control mechanisms regulating fluctuations of VT, O2, and CO2 may be different on different time scales. Thus breathing has a memory up to at least 400 breaths that can be characterized by the simple indicator alpha.